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We can define the rank of set $X$ as the least ordinal $\alpha$ such that $X \in V_\alpha$, where $V_\alpha$ is the $\alpha$-th stage of the cumulative hierarchy. However, I have yet to see this idea applied anywhere, and I spend a LOT of time on wikipedia. What are its major uses?

goblin GONE
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  • The existence of a rank function is equivalent to the axiom of regularity. http://math.stackexchange.com/questions/213639/where-is-axiom-of-regularity-actually-used/ may interest you. – Asaf Karagila Apr 27 '13 at 13:42
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    The rank is something to do induction on. That is the main (and frequent) use. – André Nicolas Apr 27 '13 at 14:23
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    Besides being a ladder for induction, it’s useful for Scott’s trick. – Brian M. Scott Apr 27 '13 at 17:32
  • Obviously related to doing induction, it's useful for definitions by recursion. You can define a property (blank) of sets by saying that a set has property (blank) if all of its elements have property (blank). This looks circular, but it's a definition by recursion on rank. – Iian Smythe Apr 27 '13 at 21:44

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